Comparing processes as curves of distributions
Counts and the table below are generated from manifest.json, so they
cannot disagree with the record. Newest finding: 2026-07-26.
| Finding | Status | Rung | Date | |
|---|---|---|---|---|
| 1 | A process is literally a curve on a sphere | active | 5 | 2026-07-21 |
| 2 | Time-averaging is quadratically blind to a brief event | provisional | 2 | 2026-07-21 |
| 3 | The information-geometry ground beats Euclidean on real speech across voices | active | 3 | 2026-07-23 |
| 4 | The same win replicates on handwriting, with no audio at all | active | 4 | 2026-07-23 |
| 5 | The method fails where identity is static — the predicted boundary | retracted | 3 | 2026-07-23 |
| 6 | The corrected boundary: two lanes, not one | active | 3 | 2026-07-24 |
| 7 | Pitch invariance by construction, not by training | active | 5 | 2026-07-22 |
| 8 | The distinctive ground does NOT generalise across light, echo and turbulence | negative | 5 | 2026-07-22 |
| 9 | The min-mean distance 'win' was a padding degeneracy | retracted | 5 | 2026-07-22 |
| 10 | Doppler / velocity measurement is not an edge | negative | 5 | 2026-07-22 |
| 11 | Against a published method: the field's warp is better, our ground still wins | active | 6 | 2026-07-23 |
| 12 | The win decomposes: trajectory > ground > warp | active | 4 | 2026-07-24 |
| 13 | The geometry needs 5–10× less data than a standard network | active | 4 | 2026-07-25 |
| 14 | It does not forget — and cannot, by construction | active | 5 | 2026-07-26 |
| 15 | A compressed latent representation does NOT help | negative | 5 | 2026-07-25 |
| 16 | Anything between the landed point and the answer must start as a pass-through | active | 5 | 2026-07-25 |
| 17 | On discrete aligned data the warp HURTS and the ground is neutral | retracted | 5 | 2026-07-25 |
| 18 | Hierarchy does not beat a flat representation | negative | 5 | 2026-07-25 |
| 19 | The geometry is not fp16-safe | retracted | 1 | 2026-07-24 |
| 20 | Measurement error: a growing label space looked like memory decaying | retracted | 2 | 2026-07-26 |
| 21 | An impossible value exposed a flaw that only a second dataset could reveal | active | 5 | 2026-07-26 |
| 22 | The Riemann work reproduces known mathematics and proves nothing new | negative | 5 | 2026-07-22 |
| 23 | Character spacing is not Riemann level repulsion | negative | 5 | 2026-07-25 |
| 24 | Memory helps a language model, but the geometry is not why | retracted | 5 | 2026-07-25 |
| 25 | The Fisher-Rao ground IS a better retrieval key than L2 -- once characters have a learned metric | active | 5 | 2026-07-26 |
| 26 | The learned character metric encodes linguistic class, and it is not frequency | active | 3 | 2026-07-26 |
| 27 | The equation predicts the next step better than a trained network, with no training | active | 5 | 2026-07-24 |
Claim
Continuing the curve along its geodesic — a closed-form step with zero learned parameters — predicts the next frame of a real signal 3-16x more accurately than a matched dense network that was trained to do it.
Evidence
Fisher-Rao error, lower is better. AUDIO (FSDD cochlea, F=32): geodesic continuation 0.0054 vs fair dense 0.0860 vs naive repeat-last 0.0866. PEN (handwriting, F=16): 0.1796 vs 0.3863 vs 0.3903. Paired: audio 646 wins / 2 losses, p=4e-190; pen 591/129, p=2e-71. The LEARNED geometry model beats fair dense on audio (p=2e-122) but only TIES on pen (397/353, p=0.12) — reported because it qualifies the headline: learning helps complex dynamics, not smooth real signals.
Scope / limits
Next-step prediction on continuous real sequences, two unrelated modalities. This is one-step continuation, NOT generation over long horizons, and not language.
Controls that could have killed it
Claim
Trained only to predict the next character, the model places characters on the simplex so that Fisher-Rao distance tracks linguistic class -- case pairs, vowels, sentence terminators, whitespace -- and this is not explained by how often the characters occur.
Evidence
Off-diagonal Fisher-Rao spread 0.5050 (min 0.1298, max 0.6349) against exactly 0.0000 for one-hot. Correlation with frequency is nil: Spearman +0.009, ~0.0% of rank variance. After residualising distance on the frequency gap, within-class pairs are closer than cross-class pairs by 0.0226; a 10,000-shuffle permutation null over class labels gives z=+4.46, p=0.00010. Tightest classes are the terminators . ? ! (-0.256) and separators , ; : (-0.212).
Scope / limits
One trained checkpoint (HODOS_embed, 30k steps, val ppl 5.29) on TinyShakespeare. Learned embeddings clustering by character class is a KNOWN property of neural language models and is not claimed as novel here. It is recorded because it is the precondition that makes finding 25 interpretable: without it there is no metric for the Fisher-Rao ground to act on.
Controls that could have killed it
Claim
Given character positions that carry a real metric, retrieving past contexts under the Hodos process distance predicts the next character better than retrieving under plain L2 over the identical frames. The only difference between the two arms is the distance function.
Evidence
Pooled over 5 seeds, 1000 held-out positions, each seed tuning its own temperature and blend weight on its own separate half: Hodos beats L2 by +0.0970 nats, p=2.0e-05. The warp HELPS by 0.0674 nats (p=2.5e-03) and the ground helps by 0.0296 nats (p=0.024) -- both reversing the retracted findings 17 and 24. The same experiment run in the old one-hot space reads -0.0249 nats, p=0.25: indistinguishable from zero, as a constant ground must be.
Scope / limits
k-NN character language modelling on TinyShakespeare against a unigram term. The learned metric is the 4,225-parameter embedding table from the trained HODOS_embed language model, so this shows the geometry works ON a learned metric -- it does not show the metric can be obtained without learning one.
Controls that could have killed it
Claim
ORIGINAL WORDING, NOW KNOWN TO BE WRONG: 'Retrieval is what helps; the geometric distance is not better than a flat one for this.'
Evidence
Memory improves by +0.105 nats, p=2.7e-06. Plain L2 retrieval beats Hodos retrieval by 0.121 nats, p=7.35e-04, and this survives per-arm temperature tuning.
Scope / limits
RETRACTED 2026-07-26. The geometry was never given a metric to work with.
Controls that could have killed it
Why this was wrong
Both of these were measured in a space where the Fisher-Rao ground is a CONSTANT FUNCTION. Contexts were built as label-smoothed one-hot rows, and on the actual vocabulary (V=65, eps=0.03) the Fisher-Rao distance between every pair of distinct characters is identical: min 2.9971, mean 2.9971, max 2.9971 -- spread exactly 0.0000. In that space Fisher-Rao and L2 induce the SAME ranking, so 'the ground ties L2' was an algebraic identity, not a measurement, and the warp was the only thing left that could move. The experiment could not have produced any other answer. Re-measured in a non-degenerate metric (finding 25) both conclusions reverse: the ground beats L2 and the warp HELPS. The H1 half of this finding -- that memory helps a language model at all -- survives and is carried into finding 25. Only the control's verdict on the geometry is withdrawn.
Claim
The apparent 'empty band' in character separations is an artifact of one-hot encoding, not a repulsion signature.
Evidence
One-hot characters have exactly ONE distinct pairwise distance — all 2080 pairs at 1.498555, spread 0.00e+00. Arbitrary points on the SAME simplex give 1334 distinct distances, so the constraint is the encoding.
Scope / limits
The structural question is genuinely shared — both ask what a distribution of separations looks like near zero. But level repulsion is a statement about a DENSE 1-D spectrum, and 65 points in 65 dimensions is the sparsest possible arrangement.
Controls that could have killed it
Claim
The distance works as a REFEREE — it correctly ranks a quantum-chaotic drum closest to the real Riemann zeros — but every attempt to use it to find a load-bearing arithmetic operator returned a null.
Evidence
Referee ranks GUE 0.0435 closest, GOE 0.0695 second, noise far. Reproduces Montgomery–Odlyzko (known). The relationship-kernel looked load-bearing at 4–6σ against random integers — but a density-matched control scored BETTER (0.062 vs 0.128), so the signal was a density-envelope artifact.
Scope / limits
RH is not proven, not approached. The real limiter is resolution: an arithmetic fingerprint needs ~10³–10⁴ zeros and we have hundreds.
Controls that could have killed it
Claim
A negative forgetting score — sequential beating joint — is impossible if the store is order-invariant, and chasing that impossibility found a real defect in the comparison.
Evidence
Handwriting reported hodos forgetting −0.083 on one seed. Cause: an incomplete final chunk is dropped from the task list, but the joint comparison arms still iterated ALL classes — so joint faced a 17-way problem while sequential faced 16-way.
Scope / limits
Speech divides evenly into pairs, so nothing was orphaned there and the flaw could ONLY surface on a second modality.
Claim
ORIGINAL WORDING, NOW KNOWN TO BE WRONG: 'the geometry forgets too — task-1 accuracy drops 0.500 as later tasks arrive.'
Evidence
The raw drop, which is real but means something else.
Scope / limits
RETRACTED same day. This is a flaw in the METRIC, not in either system.
Controls that could have killed it
Why this was wrong
In class-incremental learning the LABEL SPACE GROWS: task 1 is a 2-way choice when first learned and a 10-way choice at the end. Comparing those two numbers makes the task getting harder look like memory degrading. Measuring against the joint-trained ceiling makes the growing label space cancel, leaving only the cost of having learned in sequence — which for this system is exactly zero. The file's own docstring warned about this trap before the first version was written that way anyway.
Claim
ORIGINAL WORDING, NOW KNOWN TO BE WRONG: 'sph_log takes arccos of an inner product very close to 1.0; fp16 resolution there is coarser than the guard clamp, so every Log map collapses to zero and the geometry dies silently.'
Evidence
None — this was reasoned, never measured. Rung 1.
Scope / limits
RETRACTED 2026-07-24 by direct measurement on a T4.
Controls that could have killed it
Why this was wrong
Both halves were wrong. The premise is BACKWARDS: characters are label-smoothed one-hots, so two DIFFERENT consecutive characters are nearly ORTHOGONAL (real-text mean inner product 0.096), nowhere near 1.0. The fp16 danger band contains 0.0000 of real pairs and zero pairs lose real motion. And naive autocast never reaches the geometry anyway — the manifold tensors come out float32. The 'safety' code written to guard this imaginary hazard then introduced the only real bug the run found.
Claim
Lifting a trajectory into nested levels gives no advantage over the flat version on either modality.
Evidence
Speech: flat 0.920, lifted 0.935 — but the SMOOTHING control also 0.935, identical. Handwriting: lifted 0.870 < flat 0.877.
Scope / limits
FSDD utterances are 24 frames, so a lift yields ~5 — shallow. A null here means 'no benefit on short sequences', NOT 'no benefit'. The case hierarchy exists for is long sequences.
Controls that could have killed it
Claim
ORIGINAL WORDING, NOW KNOWN TO BE WRONG: 'For character contexts, the Fisher-Rao ground ties plain L2 exactly, and the entire performance gap is the alignment machinery.'
Evidence
Per-arm tuned: full Hodos 3.1117, ground-without-warp 2.9949, L2 2.9903. Ground vs L2: −0.0046, p=0.586 — a dead tie. The warp costs +0.1168, the whole gap.
Scope / limits
RETRACTED 2026-07-26. The tie was forced by the representation, not observed.
Controls that could have killed it
Why this was wrong
Both of these were measured in a space where the Fisher-Rao ground is a CONSTANT FUNCTION. Contexts were built as label-smoothed one-hot rows, and on the actual vocabulary (V=65, eps=0.03) the Fisher-Rao distance between every pair of distinct characters is identical: min 2.9971, mean 2.9971, max 2.9971 -- spread exactly 0.0000. In that space Fisher-Rao and L2 induce the SAME ranking, so 'the ground ties L2' was an algebraic identity, not a measurement, and the warp was the only thing left that could move. The experiment could not have produced any other answer. Re-measured in a non-degenerate metric (finding 25) both conclusions reverse: the ground beats L2 and the warp HELPS.
Claim
A randomly-initialised readout layer costs ~21 perplexity — a 2.5× degradation — regardless of its form. Initialised as a pass-through, the cost vanishes entirely.
Evidence
At identical dimension and parameters: no readout 13.528, random-init readout 34.318, identity-init readout 13.187. Both readout FORMS cost the same, so it is not which readout but having one.
Scope / limits
Applies to anything built downstream of a landed point — the memory, reasoning and decoder organs all sit there.
Controls that could have killed it
Claim
Squeezing characters onto a lower-dimensional concept simplex is monotonically worse, at matched parameters, with a working readout.
Evidence
13.545 (65 dims) → 16.089 (32) → 19.418 (16) → 23.113 (8) perplexity. No sweet spot.
Scope / limits
Character language modelling, small local config.
Controls that could have killed it
Claim
Taught classes in sequence, a standard network loses half to four-fifths of its first-task accuracy. This system loses nothing, because learning is appending to a store rather than overwriting weights.
Evidence
Forgetting measured against the joint-trained ceiling so the growing label space cancels. Speech: net +0.500, hodos +0.000. Handwriting: net +0.801, hodos +0.000. Per-seed hodos [0,0,0] on both. Learning INCREMENTALLY beats the network trained on EVERYTHING AT ONCE: 0.544 vs 0.422 (speech), 0.845 vs 0.702 (handwriting).
Scope / limits
Two modalities, 3 seeds each. Handwriting uses a random per-class holdout (no speaker axis), a weaker split than the across-speaker speech version.
Controls that could have killed it
Claim
Class-barycenter prototypes with NO training reach a standard method's 50-example accuracy using 5 examples (speech) or 10 (handwriting).
Evidence
Speech across-speaker: hodos 0.416/0.539/0.654/0.715/0.746 at 1/2/5/10/50 shots vs GRU 0.189/0.291/0.417/0.520/0.624. Handwriting: hodos 0.730→0.894 vs GRU 0.403→0.828. At ONE example the network is near chance (0.189 vs 0.10).
Scope / limits
Two modalities, 5 seeds, 200 test items, full shot sweep to 50.
Controls that could have killed it
Claim
The advantage over mean-pooling splits cleanly into three ingredients in a fixed order: using the trajectory at all, then the Fisher–Rao ground, then the alignment.
Evidence
FSDD across-speaker 5-shot, 3 seeds: FR+warp 0.657 / FR+nowarp 0.637 / L2raw+warp 0.602 / L2raw+nowarp 0.573 / mean-pool 0.483. Trajectory +9, ground +5–6, warp +2–3. Handwriting: same ordering, +25 / +3.4 / +1.8.
Scope / limits
Same ordering on a sound task AND a non-sound task, so it is not a speech artifact.
Controls that could have killed it
Claim
soft-DTW beats our hard-DTW warp — but the Fisher–Rao ground beats Euclidean under BOTH warps on BOTH modalities. The transferable contribution is the ground, and it composes with the field's better alignment.
Evidence
Handwriting z=2.75, errors 0.045→0.025 for soft-DTW over hard-DTW.
Scope / limits
Recommended distance is therefore soft-DTW + Fisher–Rao — adopt the field's warp, keep our geometry.
Controls that could have killed it
Claim
The conceptual link (Doppler is a time-warp, so the method should measure velocity) is real and elegant, but empirically purpose-built DSP measures it better.
Evidence
Matched filter is perfect (0.000 clean); Hodos ~0.13. On the velocity PROFILE — supposedly the structural edge — the Hodos warp-slope is too noisy (0.14–0.20) and loses even to a FLAT single-α baseline (0.07–0.12).
Scope / limits
Two fair benchmarks. Stopped after two rather than trying a third variant.
Controls that could have killed it
Claim
ORIGINAL WORDING, NOW KNOWN TO BE WRONG: 'min-MEAN + Fisher–Rao beats standard DTW on light by +0.21 (0.61→0.82), and it survives 6→12/class.'
Evidence
The apparent win.
Scope / limits
RETRACTED. Caught by Flouk: 'why does it track one better than the other — that doesn't make sense.'
Controls that could have killed it
Why this was wrong
min-mean degenerates into long, padded, cheap-repeat paths (path padding 0.00 for standard DTW vs 0.61 for min-mean) which makes it behave MORE like plain averaging (correlation with mean-pool 0.779 → 0.845). Light is precisely the task where averaging wins — so it 'won' for the wrong reason, and the same collapse HURT on echo where timing is the signal. The object is Marzal–Vidal's Normalized Edit Distance (1993), proven non-metric. RETIRED.
Claim
The Fisher–Rao ground shows no edge over plain L2 on light spectra, sonar range-profiles or turbulence. It ties; it does not beat.
Evidence
1-NN LOO, 3 seeds, ground ablation, noise swept to break the ceiling. At 18/class: light 0.45 vs 0.44, echo 0.73 vs 0.75, turbulence 0.62 vs 0.68. The ground earns its keep in 0 of 3 modalities.
Scope / limits
Recognition only. Does not test barycenter, morph, pooling or the physics reproduction.
Controls that could have killed it
Claim
Quotienting the process space by the global bin-shift group gives a distance between ORBITS, making recognition invariant to transposition exactly — not approximately, and with no examples required.
Evidence
Same vowel shifted 6 bins: distance collapses 61× (0.4611→0.0075) while a DIFFERENT vowel is preserved (0.2911) — a 39× separation. The invariant recogniser REPAIRS a real base-engine misclassification.
Scope / limits
Exact-zero only for an exact array roll; independently synthesised transpositions carry edge-tail truncation (D_inv ≈ 7.5e-3, not 1e-16). Pad-mode invariance is approximate.
Controls that could have killed it
Claim
The GROUND earns its keep wherever identity lives in the relative-space pattern — INCLUDING static posture. The WARP earns its keep only where identity lives in temporal motion. These are separate lanes; the earlier single-axis 'motion vs static' framing conflated them.
Evidence
With a representation that preserves relative space (distribution over TOTAL acceleration orientation), subject-disjoint sit-vs-stand = 0.758 against chance 0.500. Fisher–Rao ground contributes +5.8 points over Euclidean on a task with NO motion; the warp contributes 0, correctly.
Scope / limits
One dataset (UCI HAR). The two-lane structure is supported by findings 3, 4 and 17 jointly. PRIOR ART, added 2026-07-26: the same demarcation is already stated in the pooling literature — orderless pooling suffices for stationary or weakly nonstationary signals with little discriminative temporal structure, while order-aware pooling is needed where the cues are transient or phase-dependent. This finding independently REPRODUCES a known boundary; it does not establish a new one. It stays active because it is measured and it is true.
Controls that could have killed it
Claim
ORIGINAL WORDING, NOW KNOWN TO BE WRONG: 'Fisher–Rao does not beat Euclidean on human-activity windows because activity identity in a short window is static — exactly what the principle predicts.'
Evidence
HAR: averaging won, 0.24 vs 0.26–0.27.
Scope / limits
RETRACTED 2026-07-24. The null was OUR OWN BROKEN INPUT, not a property of the method.
Why this was wrong
The adapter fed gravity-removed acceleration reduced to a MAGNITUDE — a representation with direction deleted. Sitting and standing are both 'at rest, no direction', so under that input they were identical BY CONSTRUCTION. The method was not failing on static identity; it was succeeding on an input from which the answer had already been removed. A null produced by a crippled representation is not a boundary.
Claim
The identical claim holds on pen motion — a completely different modality — and more strongly than on speech.
Evidence
UCI Character Trajectories, 2858 real handwritten characters, 4000 triples: paired McNemar z≈4.95 (Fisher–Rao fixes 49 errors, breaks 10).
Scope / limits
Single-writer temporal recognition, not an across-body handwriting split. Not a published leaderboard.
Controls that could have killed it
Claim
Recognising the same spoken word through a DIFFERENT human voice, the Fisher–Rao ground reduces error ~17% vs Euclidean.
Evidence
FSDD, 2000 triples, fresh seed: 0.1500±0.0080 vs 0.1805±0.0086 (~2.6σ).
Scope / limits
Word-level ABX on FSDD, not the published phone-level ZeroSpeech leaderboard. Within-speaker is a TIE — the N=400 within-speaker 'win' did NOT replicate and was a small-sample mirage.
Controls that could have killed it
Claim
Averaging a signal over time loses a short event at O(ρ²) while the curve distance loses it at Θ(ρ) — so pooling destroys exactly what a process comparison keeps.
Evidence
Log-log slopes 1.90 vs 1.00.
Scope / limits
Holds under an interior/support condition (corrected by Lee's review, 2026-07-22). A novelty check found the underlying effect is KNOWN prior art in detection theory and the DTW-averaging literature — the exponent statement may be a small novel instance at most. Do NOT claim as a new theorem.
Controls that could have killed it
Claim
The map p ↦ √p sends the simplex onto the unit sphere, under which Fisher–Rao becomes ordinary great-circle geometry. A signal over time is therefore a curve on a sphere, and comparing signals is comparing curves.
Evidence
2·geodesic == fisher_rao to 1e-9; chi² ≈ ¼·d_FR² with correlation 0.987 on real audio frames.
Scope / limits
Mathematical identity, not an empirical claim. Prior art (information geometry); reused, not invented.
Every equation the project rests on, with an honest label on each: ASSEMBLED from known parts, KNOWN and merely reused, or OURS. Read ASSEMBLED carefully — it is not a lesser category. Nearly every method in this field is a composition, and §1, the Hodos distance itself, is ASSEMBLED: the contribution is which ground it is composed with, and that composition is not something we found elsewhere. What the labels buy is that nothing here can be dismissed by pointing at a part of it.
A signal becomes a process P = (pⁱ, …, pᵀ), each frame a probability distribution over F bins. Δ^{F−1} is the probability simplex. BC(p,q) = Σᵇ √(pᵇ qᵇ) is the Bhattacharyya coefficient.
ASSEMBLED — the composition IS the contribution
D(P,Q) = ( min_{π ∈ Π_β} Σ_{(i,j)∈π} g(p⁽ⁱ⁾, q⁽ʲ⁾) ) / |π★|Plain: Line the two signals up in time so the total per-step difference is smallest, then divide by the length of that alignment.
Provenance: Dynamic Time Warping (Sakoe–Chiba 1978) composed with an information-geometry ground g. The warp is standard and reused; the ground is classical; THE COMPOSITION IS THE CONTRIBUTION — an elastic distance whose local cost is the exact Fisher–Rao geodesic between distributions, with closed-form geodesic operators on top. A literature check on 2026-07-26 found the neighbouring families (OTW, TiOT, TAOT, Riemannian Time Warping) but not this composition as a working engine with these operators. D is an alignment-tolerant divergence, NOT a metric; ordinary DTW fails the triangle inequality.
Correction: An earlier version of this sheet presented the minimum-MEAN alignment as 'the distinctive object'. That is RETIRED: minimising the mean directly is the Normalized Edit Distance (Marzal–Vidal 1993), whose optimum pads the path with cheap repeats and collapses toward plain averaging. Its one apparent benchmark win was that padding artifact (finding 9).
KNOWN
chi2(p,q) = ½ Σᵇ (pᵇ − qᵇ)² / (pᵇ + qᵇ) fisher_rao(p,q) = 2·arccos( BC(p,q) ) hellinger(p,q) = √( 1 − BC(p,q) ) wasserstein1 = Σᵇ | CDF_p(b) − CDF_q(b) |
Plain: Five interchangeable ways to measure the difference between two single frames. All closed-form, none learned.
Provenance: All established — Le Cam / Topsøe, Rao 1945, Amari, Bhattacharyya, optimal transport. We reuse them; we do not claim them. Our only move here is making the ground pluggable in one engine.
KNOWN, load-bearing framing
φ(p) = √p maps the simplex onto the positive orthant of S^{F−1}
‖√p‖² = Σᵇ pᵇ = 1
geodesic(p,q) = arccos( BC(p,q) )
d_FR(p,q) = 2·arccos( BC(p,q) )Plain: Taking the square root of a distribution puts it on a sphere, and the distance between distributions becomes the plain great-circle angle. So a signal over time is a curve on a sphere, and comparing signals is comparing curves.
Provenance: Classical information geometry (Rao 1945; Amari). Known — used as the exact geometric home of the engine. It is what makes the closed-form geodesics and means below possible.
KNOWN math, our operators
slerp(p,q,t) = unembed[ ( sin((1−t)θ)·√p + sin(tθ)·√q ) / sin θ ]
karcher_mean({pᵢ}) = the point m minimising Σᵢ d_FR(m, pᵢ)²Plain: How to walk between two distributions along the shortest path, and how to average a set of them without leaving the space.
Provenance: Standard Riemannian geometry and DTW-barycenter averaging (Petitjean 2011). Known. These drive the morph and barycenter operators, and the zero-training prototypes behind findings 13 and 27.
ASSEMBLED — downgraded from OURS on 2026-07-26
D(A,B) = Θ( ρ · g(s,s′) ) — LINEAR in ρ
g(ā, b̄) = O(ρ²) under the interior/support condition
= O(ρ) otherwise — the honest boundary
so D / g(ā,b̄) = Θ(1/ρ) → ∞Plain: Averaging a signal over time is QUADRATICALLY blind to a brief event riding on an ongoing one — the final /t/ against /p/ on a sustained vowel — while the warp distance stays LINEARLY sensitive. Averaging can only lose; on brief transients it loses badly.
Provenance: DOWNGRADED after the literature check this section had been carrying a warning about since 2026-07-22. It splits into three parts and they do not have equal status. (a) The O(ρ²) half is CLASSICAL: every smooth f-divergence is locally quadratic, its second derivative IS the Fisher information, and for chi-squared the literature states χ²(P₁‖P₂) = (θ₁−θ₂)² I_F + o(·) directly. Our proof is that expansion applied to the perturbation ρu. (b) Our 'honest boundary' — the O(ρ) case where mass moves where the baseline is ≈0 — is the classical regularity condition under which the local expansion holds at all. We rediscovered the caveat that ships with the theorem. (c) The warp's Θ(ρ) half is one line from the definition, and that averaging dilutes brief events is not only known but actively engineered around (attention pooling exists for it). The COMPOSED separation Θ(1/ρ)→∞ was not found anywhere — but six web searches are weak evidence of absence, and a composition of two textbook facts is a thin novelty claim regardless. Full write-up: paper/LITERATURE-CHECK-S5.md.
Correction: This section was tagged 'OURS — the main new result' from 2026-07-22 to 2026-07-26, always carrying its own warning that its novelty was unchecked. The check was run and the warning was justified. The measured consequence (finding 2 — averaging is quadratically blind to a brief event) STANDS and is unaffected; what changes is only the claim to have discovered the mathematics behind it.
OURS — new extension
D_inv(P,Q) = min_s D(P, σ_s Q) isometry: g(σ_s p, σ_s q) = g(p, q) EXACTLY, for the full cyclic group (verified to 4.4×10⁻¹⁶)
Plain: A pitch shift is a translation along a log-frequency axis, so quotient the space by that shift. What is left is relative formant spacing — which is identity. Invariance by construction, with nothing trained.
Provenance: OURS as an extension, BUT the idea — shift-invariant and transposition-invariant distances, chroma/CQT shift, quotient and orbit metrics — already exists in music information retrieval and group theory. A specific clean instance, not a first. Measured as finding 7: a same-vowel transposition collapses 61× while a different vowel is preserved.
KNOWN — reproductions, NOT ours
R₂(r) = 1 − ( sin(πr)/(πr) )² GUE pair correlation (Montgomery 1973) H = (XP + PX)/2 Berry–Keating operator Weil/Connes explicit-formula operator
Plain: Established results run THROUGH Hodos as a referee, not equations we made.
Provenance: Everything reproduces known mathematics. RH is NOT proven. Hodos's role is the measuring instrument — it correctly ranks the GUE spectrum closest to the real zeros. Two exciting signals died to fair controls here (findings 22 and 23), both density artifacts. The honest limiter is resolution: the arithmetic fingerprint needs 10³–10⁴ zeros and we compute a few hundred.
We assembled a real, working, model-free engine — DTW plus an information-geometry (Fisher–Rao √-sphere) ground over sequences of distributions — with clean geodesic operators and a quotient extension for pitch invariance. The composition in §1 and the quotient in §6 are the parts that are ours. §5 was demoted on 2026-07-26 when the literature check it had been flagged for was finally run. The strongest results in this project are not in this tab at all — they are the MEASUREMENTS (findings 13, 14 and 27), which were built with controls designed to kill them and survived.
Every rule here was written after something went wrong. They are not general good practice; they are scar tissue.
A test whose bar is set afterwards can always be read as a success. Finding 13 predicted that the advantage would shrink with data — the signature of a prior. It grew instead. That makes it a better representation rather than a better prior: a different claim, and one we would have described wrongly had the criterion not been fixed in advance.
Findings 8, 9, 18 and 22 were all killed by a control built for that purpose. Finding 18's smoothing arm matched the result exactly, which is what ended it. A benchmark with no arm capable of producing a negative is a demonstration, not a test.
Findings 5 and 19 both passed on inputs that could not have shown the effect. The activity adapter deleted direction, so two postures were identical by construction. The fp16 test used random data that never approaches the regime in question. A null from a crippled input is not a boundary.
Finding 9 won on one modality and lost on another with no account of why. Investigating the mechanism showed it was collapsing toward plain averaging — which is exactly why it won on the task where averaging wins. Explain the mechanism before banking the win.
Finding 23 returned a statistic that was identically zero for the real data and for all 900 shuffled controls. A z of zero against a null with zero spread is a dead instrument, not a null result. It is reported as a failed measurement.
Finding 21 reported a negative forgetting score, which cannot happen if the store is order-independent. Chasing it found a real defect — and fixing it made the opposing method look worse, meaning the bug had been flattering the opponent.
Speech and handwriting share no sensor, no physics and no preprocessing. Finding 4 is what turned finding 3 from a sound trick into a claim. And per finding 21, a second dataset is not only confirmation — it is a different defect detector.
6 findings on this page are retracted, with the original wording kept and labelled. 6 more are outright negatives. Deleting them would make the record look stronger and be worth less.
| active | Survives the controls it was tested against; currently believed. |
| provisional | Measured, but on one task or at small scale; not yet replicated. |
| negative | Tested and did NOT hold. Reported because a negative is a result. |
| retracted | We claimed it, then our own measurement refuted it. Original wording kept visible. |
| untested | Stated but never measured. Carries no weight. |
| What it takes | |
|---|---|
| 1 | reasoned, unmeasured |
| 2 | measured once, synthetic data |
| 3 | measured on real data, one modality |
| 4 | replicated on a second, unrelated modality |
| 5 | survives a fair control designed to kill it |
| 6 | survives comparison against a published method |
| 7 | reproduced independently, outside this project |
A signal is a curve of probability distributions on a statistical manifold; comparison is a geodesic distance between curves.
The ground — the information-geometry representation, not the alignment machinery — is what earns a measurable advantage, and it does so wherever identity lives in the relative-space pattern of a process. The warp is a separate lane: it earns its keep only where identity lives in temporal motion, and on discrete already-aligned data it actively hurts (finding 17).
These are two lanes. Conflating them was our own error, retracted as finding 5 and corrected in finding 6.
It learns from 5–10× less data than a matched standard network, on two unrelated modalities (finding 13). At one example the network is barely above chance.
It does not forget (finding 14). Taught in sequence, a standard network loses half to four-fifths of what it first learned. This loses nothing — and cannot, because building the store in reverse order yields byte-identical answers, so the final state provably does not depend on the sequence.
It is invariant by construction rather than by training (finding 7). A transposed signal collapses 61× while a genuinely different one is preserved. A network has to learn that from examples and only ever approximates it.
The store grows. Consolidation folds repeated episodes into one prototype per concept, so it scales with how many distinct things are known rather than with how much has been seen — but lookup gets slower as it learns, where a network's does not. That is the real trade, and it is why a fast retrieval index exists.
Not a universal oracle. It solves what can be posed as a least-cost path on a distribution-manifold; it does not invent a domain's law and does not reach discrete or logical truths. The Riemann work (finding 22) reproduces known mathematics and proves nothing new.
Not better on language. Memory helps a language model, but a plain distance retrieves better than this one for text (finding 24), and the mechanism is understood (finding 17).
Not novel wholesale. Dynamic time warping, Fisher–Rao, the square-root sphere, soft-DTW, barycenter averaging and optimal transport are all prior art, reused. What is ours is the synthesis, the pitch quotient, and the measurements on this page.